A vertex creases the paper twice

A paper limits spacing, not density corrected the first bound these essays put on how finely a sheet can be creased. A crease is a band about six sheet thicknesses wide, and what a paper runs out of is room between two creases that do not meet, not metres of crease per square metre; on that reading a Miura on a 170 mm sheet of copier paper can go to 262 cells a side rather than 139.

That correction was bought with an exemption. Creases that share a vertex were allowed to come as close as they liked, because meeting is what a pattern is made of. The exemption is not free. Near the point where two creases meet, their bands overlap, and the paper there is creased twice. How far out that overlap runs is a small piece of trigonometry, and charging it to each crease gives a third bound on fineness which binds before the second.

How far two meeting bands overlap #

Take two creases leaving a vertex at an angle $\theta$, each the centre of a band $w$ wide. Walk along the first crease a distance $x$ from the vertex. If $\theta$ is at most a right angle, the nearest point of the second crease is at the foot of a perpendicular, $x\sin\theta$ away; if $\theta$ is wider, the nearest point of the second crease is the vertex itself, $x$ away. The two bands overlap wherever that distance is less than $w$, so they overlap out to

$$x^{*} = \frac{w}{\sin\theta} \quad (\theta \le 90^\circ), \qquad x^{*} = w \quad (\theta \ge 90^\circ).$$

The first figure draws that function. It is flat at one band width for any sector angle of a right angle or more, and it rises steeply below: two band widths at 30°, nearly six at 10°. A vertex with generous sectors creases the paper twice for one band width round the point; a vertex with a narrow sector creases it twice a long way out along both of the creases that bound it.

At the end of any crease there are two neighbouring creases, one on each side, and the overlap runs as far as the larger of their two reaches. So every crease end has a number, in band widths, set by the narrower of the two sectors beside it. And every crease has two ends.

A crease that is overlap from end to end #

Add the reaches at a crease’s two ends. If the crease is no longer than that sum, there is no stretch of it where the paper is creased once. It is double-creased from one vertex to the other, which is to say that the paper has no way to fold along this line independently of the creases on either side of it.

That is a sharper form of something the shortest crease is not a crease noticed. That essay found creases on a generated patch shorter than a wavelength of light, and said that a crease shorter than the width of a crease is not a crease. The trigonometry above says the threshold is not one band width but the sum of two reaches, and that the reach depends on the angle: a crease between two vertices with right-angled sectors needs to be longer than two band widths, and one flanked by 45° sectors at both ends needs more than 2.83.

On the printed shelf, with copier paper’s band of 0.6 mm, every crease has plenty of room. The tightest is on the tapered corrugation, whose sectors narrow to 65.9° toward its tapered end and whose tightest crease is 20.4 mm long: its two ends overlap for 2.10 band widths, 1.26 mm, and the pattern could be printed sixteen times smaller before that crease became all overlap. The waterbomb, the hexagon twist and the Yoshimura come next, at twenty times; the preliminary base, with eight long creases, could shrink eighty-eight times.

The ranking is not the ranking by density, and it is not the ranking by closest spacing either. The preliminary base shares the narrowest sectors on the shelf, 45°, with the waterbomb, and has the most room, because its creases are half the sheet long. The Yoshimura has 60° sectors and short creases. What the vertex bound measures is the ratio of a pattern’s shortest crease to the overlap its sharpest sectors put at the ends of it — two quantities that no density or spacing reads.

The Miura, charged for its vertices #

A Miura vertex has sectors a little either side of a right angle; on the family drawn here the narrow ones are 69.9°. At that angle the overlap runs $1/\sin 69.9^\circ = 1.065$ band widths along each crease, and a crease with such sectors at both ends carries 2.13 band widths of overlap. The shortest creases with those sectors at both ends are the straight ones, a cell long.

So one cell of straight crease becomes all overlap when a cell is 2.13 band widths long. Measured over every crease of a Miura twenty-four cells a side, the tightest has 0.477 of a cell of length for each band width of overlap at its ends, and a 170 mm sheet reaches that when

$$n = \frac{0.477 \times 170\ \text{mm}}{w}.$$

On copier paper that is 135 cells a side. The spacing bound gave 262.

The vertex bound is about half the spacing bound on every paper — 135 against 262 on copier paper, 193 against 374 on kami, 337 against 655 on washi — because both scale as one over the band width and their ratio is a constant of the pattern: 0.477 against 0.925, the length a cell has beyond its vertices’ overlap against the gap between two creases that do not meet.

The waterbomb, charged the same way #

The waterbomb’s vertices are degree six and eight with sectors of 45°, so its reaches are longer: 1.41 band widths at a 45° end. Its tightest crease is a half-diagonal, 0.707 of a cell long, with 2.41 band widths of overlap between its two ends — one end flanked by 45° sectors, the other by wider ones. That is 0.293 of a cell for each band width, and on copier paper it gives 82 cells a side.

Against its spacing bound of 141 that is 58 per cent. The waterbomb’s closest non-meeting creases are half a cell apart, so it was already the family that meets the paper sooner by spacing; charged for its vertices it meets the paper sooner again, and by a larger share than the Miura, because its narrow sectors push the overlap further out along creases that were short to begin with.

Three bounds, and the one that was nearly right #

Set the three bounds side by side for copier paper and a pattern emerges that neither of the earlier two essays could see:

The density bound had the wrong argument and nearly the right number. It assumed a second family of creases halves the spacing, which is false; it put the finest Miura at 139 cells and the finest waterbomb at 74. The vertex bound, which makes no such assumption and charges the overlap near each vertex instead, puts them at 135 and 82 — three per cent below for the Miura and eleven above for the waterbomb.

That agreement deserves suspicion rather than relief. For the Miura, a cell carries about two cell-widths of crease, so the density bound is roughly half a cell per band width, and the vertex bound is 0.477 because a straight crease a cell long carries a little over two band widths of overlap. For the waterbomb, a cell carries 3.8 cell-widths, so density gives 0.26, and the vertex bound is 0.293 because a 0.707 half-diagonal carries 2.41. Two unrelated ratios that happen to land within a tenth of each other on two families is an observation, not a law, and a family with long creases and narrow sectors would pull them apart.

What the comparison does establish is the order. On both families the spacing bound is the loosest by nearly a factor of two, and the bound that binds is set at the vertices, where the earlier correction had declared the ground free.

What a paper can change, and what it cannot #

All three bounds are a constant of the pattern divided by the band width, so a thinner paper moves them together and by the same factor. Foil-backed tissue at 26 µm has a band 3.85 times narrower than copier paper’s, and it takes the Miura’s vertex bound from 135 cells a side to 519 and the waterbomb’s from 82 to 319 — each exactly 3.85 times further, to the rounding of a cell. Nothing about the paper changes which bound binds or which crease meets it first. The order of the three bounds, and the identity of the crease that runs out of room, are properties of the drawing.

So the only lever on the vertex bound that is not a material is the pattern’s own geometry, and the Miura has one. Its sectors are set by the lean of its zigzag: the more it leans, the narrower the narrow sectors and the further the overlap runs. With no lean at all the sectors are right angles, the overlap at each end is one band width, and a straight crease one cell long becomes all overlap at exactly half a cell per band width — the most a Miura cell of that length can manage. The family drawn here, at a lean giving 69.9° sectors, reaches 0.477.

The lean costs the Miura under five per cent of its fineness. That is a small price for the property the lean exists to buy — a zigzag, rather than a flat grid, is what gives the Miura its single folding motion — and it is a price that can now be read off one sine rather than discovered at the paper.

The waterbomb has no such lever. Its 45° sectors are fixed by the geometry of a square cell cut by both diagonals, so its half-diagonal will always carry 1.41 band widths of overlap at its sharp end, and its 0.293 of a cell per band width is a constant of the pattern rather than of any choice made in drawing it. A designer who wants a waterbomb finer on a given paper can only choose a thinner paper.

What charging a vertex assumes #

A band is a strip of fixed width centred on the crease, the same six sheet thicknesses as before. Near a vertex a real crease is not a strip: the paper there forms a small cone or a dimple, and the crease has a radius is the fuller account of the region a fold occupies. The overlap computed here is the region where two idealised strips cover the same paper, which is where the real material has to do something no strip describes.

Overlap is charged only between neighbouring creases at a vertex. Two creases two sectors apart overlap too, but always less far than the neighbour between them, so the larger of the two neighbours’ reaches is the whole charge.

A crease that is overlap from end to end is treated as the failure. It is a threshold, not a model of what the paper does past it, and a crease slightly longer than its overlap still has only a short stretch of single-creased paper — a sliver no folder could place a fold along independently.

What the bound does not show #

It is silent about the shape of the vertex. It also treats every vertex as a point, when a real vertex in folded paper has already been blunted by the overlap it creates, so the reach computed for a crease’s second neighbour is measured from a point the material has rounded away. The overlap regions round a degree-eight vertex such as the preliminary base’s centre are eight wedges of double-creased paper meeting at the point, and at the point itself every band overlaps every other; how many layers of creasing that is, and whether a sheet can carry it at all, is a question about the material that the geometry here only frames.

It says where the first crease runs out of room, not how many do. On a family every cell repeats the tightest crease, so the first is all of them; on the tapered corrugation the tightest is at the narrow end only, and most of the pattern has room to spare.

And like the other two bounds it is a bound on a drawing. How much line is on the paper observed that the last crease laid into a sheet is not laid the way the first was, and a pattern at 135 cells a side would defeat any hand long before its vertices’ bands met.

Where the width meets the angle #

The result joins two things that have been measured separately for a long time.

Sector angles are what the flat-folding conditions are written in. Kawasaki’s theorem sums them alternately and the same vertex, found four times finds the conditions restrictive enough that a few vertex types recur everywhere. Nothing in that account has a length in it.

Band width is what the material contributes. The sheet has a thickness is where it enters these essays, and nothing in a body folds on a line is where it becomes a hinge region with a width of its own. Nothing in that account has an angle in it.

The overlap reach $w/\sin\theta$ is both at once, and it makes a flat-foldability choice into a material cost. A designer choosing a vertex with a 45° sector rather than a 70° one has chosen, without being told, to double-crease a third more paper along both of that sector’s creases — and on a fine enough sheet, to lose the crease between two such vertices altogether.

Still open: what the double-creased ground holds #

The bound counts the overlap as lost ground and asks when a crease has none left. It does not ask what the overlap region can carry, and that is the question the preliminary base’s centre forces.

Near a vertex of degree $d$ there are $d$ overlap wedges, and at the point all $d$ bands coincide. A disc one band wide round a degree-four vertex holds two band widths of crease and one round a degree-eight vertex holds four. Whether a paper can fold a vertex whose bands overlap four deep, or whether the centre of a preliminary base is always a small hole, a bulge or a tear in real material, is a measurement of paper rather than of patterns — but the geometric half, how deep the overlap is at each distance from the vertex, is computable for every vertex on the shelf.

The other direction is design. The vertex bound depends on the shortest crease and its flanking sectors, and both are choices. A Miura drawn with a less extreme lean has wider sectors and a smaller overlap per crease, so it could be folded finer by the vertex bound at the cost of a shallower zigzag; where the length sits already shows that patterns distribute their length very differently, and the vertex bound gives a reason to prefer one distribution over another that has nothing to do with the folded shape.

Both directions start from the same table, which is the shelf read by angle rather than by length, and both could be run on the generated patches as easily as on the printed ones.

The habit worth carrying is about exemptions. When an argument sets some cases aside as free, compute what they cost before trusting the bound that remains. Meeting creases looked exempt from a limit on how close creases may come, because meeting is the definition of a vertex; they were exempt from the limit and not from the material, and the material charged them at the angle.